On Maximal Layers Of Random Orders

ON MAXIMAL LAYERS OF RANDOM ORDERS Indranil Banerjee George Mason University, 2015 Thesis Director: Dr. Dana Richards In this thesis we investigate the maximal layers of random partial orders. Main contributions are two-fold. In the first half we investigate the expected size of different maximal layers of a random partial order. In particular when the points are in a plane, we give an enumerative formula for the distribution of the size of these maximal sets. Using MonteCarlo based simulation we extrapolate the results for higher dimensions. In the second part we explore the computational aspect of the problem. To this end we propose a randomized algorithm for computing the maximal layers and analyze its expected runtime. We show that the expected runtime of our proposed algorithm is bounded by o(kn2) when k is fixed and in the worst case by O(kn2). This is the first non-trivial algorithm whose run-time remains polynomial whenever k is bounded by some polynomial in n while remaining subquadratic in n for constant k. We also extend these results to random orders with arbitrary distributions. Chapter 1: Introduction Random partial orders, soon to be defined formally, are one of the more fundamental objects in combinatorics as well as in computational geometry. Unlike their much studied cousins, random graphs, random partial orders has not been studied that extensively. This is partly due to the fact that the partial ordering among elements makes independence assumptions in probabilistic analysis oftentimes elusive. Before formally introducing random orders, we shall first introduce the concept of maximal sets of partial orders. A set P along with a relation R ⊂ P × P such that R is antisymmetric, transitive and reflexive is called a partially ordered set (poset). If for all x, y ∈ P either (x, y) or (y, x) ∈ R then it is a total order (ordering between each pair of element is known). We will first introduce maximal sets in the geometric setting and then for partial orders. Let P (with |P | = n) be a set of points in some k-dimensional space such that in each dimension points are orderable[1]. When k = 2, we can view P as a set of points in a plane. Given such a set of points we often want to determine the boundary of the set. In one dimensional space, when P is just points on a line 1 the boundary set can be defined unambiguously. That is, it consists of the minimum and the maximum point of P . If P is a multi-set then there may be more than one maximum or minimum point or both. However, we can define the boundary set in several ways if P resides in a plane or in some higher dimension. When P consists of points on a plane one of the most common way to define the boundary of P as the smallest convex set Q ⊂ P , such that if a convex set contain Q then it contains P . This convex set Q is called the convex hull CH(P ) of P . In the plane, the shape of this set is just a convex polygon. For any P the convex hull CH(P ) is unique modulo points that are identical. We shall assume the finitary setting here: that is objects are finite as well as the dimensions in which they reside. This makes the algorithmic aspect of our study meaningful. 1 In this thesis we look at a different kind of boundary set, called the maximal set of P . Before we can introduce it, we need to define some terms first. Let P be a set of k-dimensional points. We shall use x[i] to denote the ith coordinate of x. Definition 1.1. For two points x, y ∈ P we say x y (x dominates y or x is above y) if x[i] ≥ y[i] ∀i ∈ [1...k]. Definition 1.2. If neither x y nor y x then we say that x and y are incomparable, denoted by x‖y. Definition 1.3. If ML(P ) is the maximal set of P then: ML(P ) = {x ∈ P | 6 ∃y ∈ P and y x} We refer the maximal set and maximal layer interchangeably. Unlike CH(P ), ML(P ) is not unique. It depends on how we define dominance. For example if instead use ≤ in place of ≥ we still get a valid definition. If defined in this way the set is called the minimal set of P . In general, we can define the extremal set of P with respect to a orthant in k-space: Definition 1.4. If ML(P,O) is the extremal set of P with respect to a orthant O, then: ML(P,O) = {x ∈ P | 6 ∃y ∈ P and y O x} and x O y if x[i] op y[i] for each i ∈ [1...k]. Where, op is either ≥ or ≤ depending on the orthant O. Thus there are 2k different extremal sets corresponding to each orthants. In this study we will not use this generalized definition but work with the one defined earlier. The results presented here carries over to this general setting without much difficulty. Next we define the ith maximal set of P : Definition 1.5. 1. ML1(P ) =ML(P ) 2